Problem 86 - Prealgebra

Question

Estimating The quotient \(1,000 \div-47\) is closer to which of the following? a. 5 b. \(-10\) c. 15 d. \(-20\)

Step-by-Step Solution

Answer

The quotient is closest to option d. (-20).

Step 1: Understand the problem

We need to estimate the result of dividing 1000 by -47 and determine which option the quotient is closest to.

Step 2: Consider magnitude of numbers

Focus on the magnitudes rather than the signs to estimate the order of the quotient. Think of 1000 divided by approximately 50 as a close estimate to understand the magnitude.

Step 3: Estimate quotient using approximate values

Divide 1000 by 50 to get an approximation: \(1000 \div 50 = 20\). This is purely considering the magnitudes.

Step 4: Account for the sign in the division

Since 47 is negative, the actual quotient will also be negative. Hence, the estimated quotient calculated as 20 should actually be \(-20\).

Step 5: Match estimated quotient to choices

Check the estimated negative quotient, \(-20\), against the provided options. The choice \(\text{d. } -20\) matches closely.

Key Concepts

Dividing Negative Numbers

When dividing numbers, it's crucial to understand how negative numbers change the outcome. Negative numbers flip the sign of the division result. This means that when you divide a positive number by a negative number, or vice versa, the quotient will be negative.

• Positive ÷ Negative = Negative
• Negative ÷ Positive = Negative
• Negative ÷ Negative = Positive

In our exercise, we have 1,000 divided by -47. Here, 1,000 is positive, and -47 is negative. Therefore, the quotient will be negative. Whenever you're handling division involving negative numbers, remember this rule of flipping the sign. It's the key to getting the sign of the quotient right.

Approximation

Approximation is a helpful mathematical strategy used to simplify complex calculations by finding a "close enough" value. When dealing with division, especially with awkward numbers, approximation can significantly simplify the process.
For the problem of estimating 1,000 ÷ -47, consider the numbers' size, or magnitude, rather than worrying about precise exact figures.

• Round 47 to the nearest feasible number for simple division—in this case, 50.
• This makes it easier to estimate that 1,000 ÷ 50 results in 20.

Use approximate values when an exact calculation isn’t necessary, saving time and effort while still providing a reasonable estimate of the result.

Integer Division

Integer division involves dividing numbers and managing the result as a whole number. This type of division truncates or ignores any remainder or decimal places, providing a straightforward whole number result.
When estimating the quotient for 1,000 ÷ -47, focus on integer outcomes. Since 1,000 isn’t perfectly divisible by 47, approximating using integer division helps give a useful estimate.

• Estimate by reshaping the division into 1,000 ÷ 50, which equals 20.
• Keep integers in mind even when the exact division might offer more decimals or a remainder.

The purpose of integer division is to simplify and provide a practical whole number output, improving ease of understanding and execution in many scenarios.